DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 397 



In (4) the separation is 1767'09 = i/ t 10'81. If it corresponds to a real ^ the limit 

 will be (5) D ( ) which reduces vj by 107 and makes the limit about 212 less. The 

 mantissa would be 82A 2 +7<5 + 71. This difference (71) from an exact multiple of J, 

 shows this to be impossible. 



The discussion of the 1864 series has definitely shown it to be of the F type, and 

 has given the limit within very small errors. This limit is one of the d ( 1 ) sequents 

 of the diffuse series. Its mantissa was found to be 81A 2 2%S or80A a +15^. It 

 gives one firm starting point for the discovery of the D series. The results just 

 obtained indicate the lines which through their dependence on multiples of A 2 give 

 the origin term of the diffuse sequence. They point, as we have seen, to the existence 

 of several displaced, or parallel, sets of diffuse series. It is possible to show definitely 

 that these exist, even if there be some uncertainty as to the lines occupying the 

 position of D n (l). In the normal case with a single diffuse series, the D u (l) is always 

 the strongest line of the series. Also in the normal type the D limit is the same as that of 

 S here 51025'29 + When however displacement occurs, the energy of a single line is 

 dispersed amongst several others, and a line corresponding to the normal may be weak, or 

 even too faint to have been observed. As a matter of experience also it is found that the 

 lines of low order (m = 1, 2...) are subject to these displacements in a much greater 

 degree than those for higher orders of m. Now there are a number of lines, which by 

 their position and absence of v l separations to stronger lines have the appearance of 

 being D n (l) lines. If they are, their mantissa? must differ from multiples of A 2 (in 

 the present case 80A 2 ) by multiples of the oun. The fact that they may do so does not 

 of course prove that they are D n lines. If they do not do so it proves that they are 

 not. They may however in the latter case belong to a displaced series, satisfying the 

 multiple law when the proper displaced limit (y$i) S (<) is employed. This gives us a 

 method of testing as to what displacement a given line may correspond. If our 

 calculus were already fully established the next step would be to apply this test to 

 the above lines. But in reality we are testing our calculus to see if it can be firmly 

 established, and our immediate aim must be to obtain independent evidence for the 

 existence of parallel series. For this immediate purpose it will only be necessary to 

 apply the test to two lines, the general question being postponed for the present. 



In the first attempt at arranging the D u series the strong lines (8) 20559'08 and 

 (10)38366'3G were taken for m = 1, 2, and the formula calculated with the limit 

 D ( QO ) = S ( oo ). As will be seen immediately, this gave satisfactory agreement with 

 sounded observed lines up to m = 15 ; and as a matter of fact this series was used to 

 test for the parallel sets displaced (2^) S ( co ) on either side of it. Now the formula 

 constants for a set only vary slightly if the wave-number of the line chosen for m = I 

 is changed considerably. This therefore did not prove definitely that 20559 is the 

 correct D n (l), and as a fact it does not satisfy the multiple test. Its mantissa is 

 897337 31'14 That of the line 19989, which is shown above to be the origin of the 

 normal D set is 879853-30'29 The difference is 17484 = 28&J+71. This is as far 



